I'm trying to do problem 7.1.7 in Dummit and Foote.
Problem: Prove that the centre of a ring $R$ is a subring of $R$ that contains the identity. Prove that the center of a division ring is a field.
I've already shown that $Z(R)$ is a subring, but my issue here is that I would like to show that if $S \subseteq R$ is a subring of a division field then $S$ must itself be a division ring. This would then show that the centre of a division ring is a field since it will be a commutative division ring. However I don't seem to be able to finish it off even though it seems basic. I know that an additive subgroup of $R$ that is also closed under multiplication and contains the multiplicative identity, but I can't seem to leverage this to show that any element in $S$ has a multiplicative inverse. As I'm sure it's basic a hint would be appreciated, thanks in advance for your help.